British Journal of Mathematics & Computer Science
9(6): 537-558, 2015, Article no.BJMCS.2015.223
ISSN: 2231-0851
SCIENCEDOMAIN international www.sciencedomain.org
Parameters Determination for the Design of Bevel Gears Using Computer Aided Design (Bevel CAD)
B. O. Akinnuli1, O. O. Agboola1* and P. P. Ikubanni1
1Department of Mechanical Engineering, Landmark University, OmuAran, Kwara State, Nigeria.
Article Information
DOI: 10.9734/BJMCS/2015/18411 Editor(s): (1) Dariusz Jacek Jakóbczak, Chair of Computer Science and Management in this department, Technical University of Koszalin, Poland. Reviewers: (1) Anonymous, Duzce University, Turkey. (2) Wang yong, Mechanical Department, TianJin University of Commerce, China. Complete Peer review History: http://sciencedomain.org/review-history/9852
Original Research Article Received: 20 April 2015 Accepted: 12 May 2015 Published: 18 June 2015 ______
Abstract
This paper prescribes a Computer Aided approach to the design of bevel gears. The approach utilizes standard design equations and standard data on bevel gears; linking them together using a Programming language(C#) to develop this special software (Bevel CAD) that designs and determines the strengths and dimensions of bevel gears. This study reviews the Procedural steps (algorithms) involved in the design of bevel gears and the development of the software package (Bevel CAD) which is to be used in designing bevel gears. When material required are selected based on the area of application, the software will make use of the data provided using the C# to determine the required; speed and velocity of the bevel gear, number of teeth, speed ratio, dynamic load, endurance strength and maximum wear load for the design bevel gear. The Bevel CAD’s performance was verified by comparing the results of Algorithm calculation and the software’s results. The Bevel CAD was confirmed effective as the minor differences obtained between the results were due to approximation errors. The Bevel CAD increases productivity but reduces drudgery of enormous calculations; hence, making it a recommended tool for industries and tertiary institutions for the designing of bevel gears.
Keywords: Bevel gear; design parameters; software development; CAD.
1 Introduction
A machine is the combination of various elements to perform a task. In doing so, power (torque) is needed to be transferred from one element or part to another; and this can be achieved with the use of power transmitting devices such as belt drives, chain drives, gear drives. Gears lead other power transmitting devices (chain drives, belts drives, etc) because of its ability to achieve definite velocity ratio. In precision
______*Corresponding author: [email protected];
Akinnuli et al.; BJMCS, 9(6): 537-558, 2015; Article no.BJMCS.2015.223
machines (clock), in which a definite velocity ratio is of importance, gears and other toothed wheels could be used.
A gear is a rotating machine element having cut teeth (or cogs) which mesh with another toothed part in order to transmit torque. Geared devices can change the speed, torque, and direction of a power source. According to [1], gear is one of the most important devices used in many types of machinery. Gears allow the user to translate power, motion and torque. Gears have a power transmission efficiency of up to 98% and are some of the most durable torque transmitting machine elements.
A bevel gear system is used for transmitting power at a constant velocity ratio between two shafts whose axes intersect at a certain angle. Gear can be used to transmit large power (small car can run on belt drives but large vehicles can’t, instead gears are being used). Gears are mostly applicable to small centre distances of shafts; they possess high efficiencies; they have reliable services; gears have compact layout. These among other characteristics of gears give it edge over other power transmitting devices. Bevel gears among other gears are used for transferring torque with intersecting shafts in the same plane.
While bevel-gear CAD software is a computer software that aids engineers, bevel-gear designers, manufacturers in the design of bevel gears. Since the World is turning towards computer technology and the use of computer to alleviate human efforts; the application of computer to the engineering world is not left behind. Hence, numerous works has been done on Computer Aided Design. Gear as an important machine component has also benefitted from this technological development. Works had also been done on gear design software and improvements are being achieved on daily basis. Therefore, [2] stated that model design of the bevel and hypoid gear is an area of great research interest nowadays. Recently, modeling of spiral bevel and hypoid gear made a lot of achievement, summarized as: 1) Point-to- surface modeling by fitting discrete points on the tooth surface, during which the vital steps are the solution of the discrete points on the tooth surface and their obtainment based on derivation of the tooth surface equations. 2) line-to-surface modeling by fitting tooth profile curves, whose core areas are the equation derivations of the tooth profile curves and the output in the three-dimensional graphics software [3].
Computer Aided Design CAD was defined by [4] as any design activity that involves the use of computer to create, modify and document engineering design using interactive computer graphics systems referred to as a CAD system. Since its inception, CAD has undergone continuous development. [5] reported that by the 1970’s, CAD systems were being used in many drafting application and by the mid-1970s, there was an established market for CAD. CAD has grown from a narrow activity and concept to a methodology of design activities that include computer or group of computers used to assist in the analysis, development and drawing of product components. One of the areas where CAD was employed in manufacturing technology as related to gear design is the Gear skiving where CAD simulation approach was used [6].
Also, CAD surface design can be used for rapid prototyping by sweeping or rotating, and creating the curves surface from the point cloud, a variety of new modeling and their respective optimization methods were explored. This can then compensate for some deficiencies in the previous modelings, and create new conditions for fast and accurate parametric modeling of the spiral bevel and hypoid gear [7].
2 Bevel Gear Review
As reported by [8], the first primitive gears can be traced back to over 3000 years ago where early gears were made from wood. They were made of wood and had teeth of engaged pins. Early Greeks used metal gears with wedge shaped teeth; Romans used gears in their mills; stone gears were used in Sweden in the Middle Ages (Gears Manufacturers). All of these cultures found reasons to use basic gearing to convert energy or motion in one form to a form they could use in devices for the technological advancement of their societies. Gears have existed since the invention of rotating machinery. However due to their force multiplying proportions, early engineers used them for hoisting heavy loads such as building materials. The mechanical
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advantage of gears was also used for ship anchor hoist and catapults. Bevel gears are indispensable parts of drive systems found in power transmission for various machinery and equipment [9]. Due to a relatively complex geometry, continuous efforts are made to streamline the design and manufacturing process of bevel gears [10]. As a result of this, there is need for accurate modeling of spiral bevel which gives room for digitized manufacturing such as the tooth contact analysis (TCA) technology, the error correction of tooth surface technology and other key technologies [11].
According to [12], mating gear teeth acting against each other to produce rotary motion may be likened to a cam and follower. As two gears mesh, there is tendency for unwanted noise to be generated. [13] claimed that accurate tooth surface and good surface quality are critical to achieve the low-noise bevel gear drives.
Bevel gear has the numerous applications in engineering and machine design. It has several importance and applications. Among these applications are:
2.1 The Differential Drives
This can transmit power to two axles spinning at different speeds, such as those on a cornering automobile. The Fig. 1 below shows a typical example of a differential bevel gear.
Fig. 1. Hypoid bevel gears in a car differential Source: [14] 2.2 Dividing Head
Dividing head used machining operations like milling gears, sprockets on milling machine.
2.3 Hand Drill Mechanism
Hand drill mechanism as the handle of the drill is turned in a vertical direction, the bevel gears change the rotation of the chuck to a horizontal rotation.
2.4 Rotorcraft Drive Systems
Spiral bevel gear are important components on this system and are used to operate at high load and high speed. In this application, spiral bevel gears are used to redirect the shaft from the horizontal gas turbine engine to the vertical rotor.
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Over the years, engineers have been coming out with various designs of bevel gears; trying to improve on their performances under various loads. This has assisted in the invention and production of machines with higher performances and efficiencies.
The bevel CAD software in discussion is a specialized software not meant for the designing of other gears or machine part unlike other multi-purpose software such AUTOCAD, INVENTOR, CATIA, PRO E. The software is mainly for the design of bevel gears. Its specialization has made this software easier and more flexible for bevel gear manufacturers and designers.
This paper aims at identifying the designed parameters required in designing bevel gear, collect relevant data for bevel gear design, develop algorithm for the steps required for the computation for each parameter, develop a software for implementing the algorithm developed and test the performance and effectiveness of the software model developed.
The numerous applications of bevel gears in machines have made its design more frequent and important. As a result of its immense importance, several works are being done on improving the efficiencies of bevel gears under static and dynamic load. This among other under-listed reasons justifies this research work:
i. Time and efforts in designing Bevel gear can be far reduced by using the BEVEL CAD software in the design. This will also assist designer focus on conception of ideas. ii. The modeled software is more flexible and user friendly; hence, making it comprehensive and easier for designer to use. iii. The software uses standard data and formula from renowned Engineering Textbooks; thus, making its result more reliable. iv. The computational speed of the modeled software is comparatively better than other multipurpose CAD software such as INVENTOR, AUTOCAD etc
3 Methodology
3.1 Terminologies and Design Parameters Identification
The design of a bevel gear starts from getting acquainted with various terminologies, symbols/parameters and formulae attributed to Bevel gear. A bevel gear is shaped like a section of a cone. See Fig. 2.0 for the nomenclature of a meshed bevel gear.
Fig. 2.0. Nomenclature of a meshed bevel gear Source: [15]
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The parameter/symbol used, description as well as the formula (as applicable) are tabulated as shown in Table 1.
Table 1.0. Parameters/symbols, description and formula used
Symbol/Parameter Description/Meaning (unit) Formula σo1 Allowable static stress of pinion wheel To be determined from the (N/mm2) material used 2 σo2 Allowable static stress of gear wheel (N/mm ) To be determined from the material used T1 or T2 Number of Teeth in pinion and wheel One of the two would be given respectively N1 or N2 Speed of pinion and gear respectively (rpm) One of the two would be given G Velocity ratio or input gear ratio (unitless) To be given o Θs Shaft angle ( ) To be given θ Pitch angle for pinion wheel (o) p1 + o o θp2 Pitch angle for gear wheel ( ) θp2 = 90 - θp1 m Module (unitless) To be determined from the AGMA standard table for tooth form D1 or D2 Diameter of pinion and wheel respectively mT1 and mT2 where m is the (mm) module de Dedendum (mm) To be determined from the AGMA standard table for tooth form a Addendum (mm) To be determined from the AGMA standard table for tooth form Td Tooth depth (mm) To be determined from the AGMA standard table for tooth form rf Fillet radius (mm) To be determined from the AGMA standard table for tooth form Tth Tooth thickness (mm) To be determined from the AGMA standard table for tooth form DR1 or DR2 Root diameter of pinion and gear respectively D1 – 2de and D2 – 2de (mm) L Slant height of Pitch cone(mm) + 2 2 V Velocity of pinion or gear (m/s) 1 or 2 60000
3.2 Gear Teeth System
A tooth system is a standard given by the American Gear Manufacturers Association (AGMA) and the American National Institute (ANSI). It specifies relationship between addendum, dedendum, working depth, tooth thickness and pressure angle to attain interchangeability of gears of a tooth number but of the same pressure angle and pitch. Gear designers should be informed of advantages and disadvantages of the various
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systems so that optimum tooth for a given design can be chosen. Basically there are four systems of gear teeth.
i 14.5º Composite system ii 14.5º full involute system iii 20º full depth involute system iv 20º stub involute system
Table 2.0. Standard proportion of gear system by AGMA
S/N 14.5º Composite 20º full depth 20º stub involute or full depth involute system system involute system 1. Addedum 1 m 1 m 0.8 m 2. Dedendum 2.25 m 1.25 m 1 m 3. Working depth 2 m 2 m 1.6 m 4. Minimum total depth 2.25 m 2.25 m 1.80 m 5. Tooth thickness 1.5708 m 1.5708 m 1.5708 m 6. Minimum clearance 0.25 m 0.25 m 0.20 m 7. Fillet radius at root 0.4 m 0.4 m 0.4 m Source: Khurmi and Gupta (2005), note: m in the Table 2.0 stands for module
3.3 Lewis, Buckingham and Tregold Formula
These equations are applied to determine the tooth form, Bevelling factor, Tooth form factor and static strength. Static strength of bevel gear is obtained by determining the following parameters.
(a) Velocity factor Cv
Velocity factor Cv is determined first base on the level of cut of the gear.
C = (For a carefully cut gear) (1) v
C = (For teeth cut by form cutter) (2) v
C = (For teeth generated with precision machines) (3) v where v is the velocity in m/s)
(b) Equivalent Number of Teeth TE
Equivalent number of teeth for both Pinion and Gear wheel are obtained from the relations shown below respectively.
TE1 = T1secθp1 (4a)
TE2 = T2secθp2 (4b)
(c) Tooth form factor Y’